The Heinz type inequality, Bloch type theorem and Lipschitz characteristic of polyharmonic mappings

Suppose that $f$ satisfies the following: $(1)$ the polyharmonic equation ${\mathrm{\Delta }}^{m}f=\mathrm{\Delta }({\mathrm{\Delta }}^{m-1}f)$$={\phi }_m$ $(\varphi_{m}\in $\mathcal{C}$(\overline{$\mathbb{B}$^{n}},$\mathbb{R}$^{n}))$, (2) the boundary conditions ${\mathrm{\Delta }}^{0}f={\phi }_0,{\mathrm{\Delta }}^{1}f={\phi }_1,\dots ,{\mathrm{\Delta }}^{m-1}f={\phi }_{m-1}$ on ${\mathbb{S}}^{n-1}$ ($\varphi_{j}\in $\mathcal{C}$($\mathbb{S}$^{n-1},$\mathbb{R}$^{n})$for$j\in{0,1,\ldots,m-1}$and$$\mathbb{S}$^{n-1}$denotestheboundaryoftheunitball$$\mathbb{B}$^{n}$),and$(3)$ $f(0)=0$, where $n\ge 3$ and $m\ge 1$ are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in \cite{CP-Hi}. Additionally, we show that if $f$ is a $K$-quasiconformal self-mapping of ${\mathbb{B}}^n$ satisfying the above polyharmonic equation, then $f$ is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as $K\to 1^{+}$ and $\mathrm{\parallel }{\phi }_j{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0^{+}$ for $j\in \{1,\dots ,m\}$.